A333491 - OEIS

%I #4 May 16 2020 14:28:50

%S 3,4,5,6,7,8,9,10,11,12,13,16,17,18,19,20,21,22,23,24,25,26,27,28,29,

%T 30,31,32,33,34,37,40,41,42,43,44,47,48,49,50,51,52,56,57,58,59,60,61,

%U 62,63,64,65,66,67,68,69,70,71,74,75,76,77,78,79,80,81,82

%N First index of partially unequal prime quartets.

%C Let g(i) = prime(i + 1) - prime(i). These are numbers k such that g(k) != g(k + 1) != g(k + 2), but we may have g(k) = g(k + 2).

%e The first 10 partially unequal prime quartets:

%e    5  7 11 13

%e    7 11 13 17

%e   11 13 17 19

%e   13 17 19 23

%e   17 19 23 29

%e   19 23 29 31

%e   23 29 31 37

%e   29 31 37 41

%e   31 37 41 43

%e   37 41 43 47

%t ReplaceList[Array[Prime,100],{___,x_,y_,z_,t_,___}/;y-x!=z-y&&z-y!=t-z:>PrimePi[x]]

%Y Primes are A000040.

%Y Prime gaps are A001223.

%Y Second prime gaps are A036263.

%Y Indices of unequal rows of A066099 are A233564.

%Y Lengths of maximal anti-runs of prime gaps are A333216.

%Y Lengths of maximal runs of prime gaps are A333254.

%Y Maximal anti-runs in standard compositions are counted by A333381.

%Y Indices of anti-run rows of A066099 are A333489.

%Y Strictly decreasing prime quartets are A054804.

%Y Strictly increasing prime quartets are A054819.

%Y Equal prime quartets are A090832.

%Y Weakly increasing prime quartets are A333383.

%Y Weakly decreasing prime quartets are A333488.

%Y Unequal prime quartets are A333490.

%Y Partially unequal prime quartets are A333491 (this sequence).

%Y Positions of adjacent equal prime gaps are A064113.

%Y Positions of strict ascents in prime gaps are A258025.

%Y Positions of strict descents in prime gaps are A258026.

%Y Positions of adjacent unequal prime gaps are A333214.

%Y Positions of weak ascents in prime gaps are A333230.

%Y Positions of weak descents in prime gaps are A333231.

%Y Cf. A006560, A031217, A054800, A059044, A084758, A089180, A124767, A333215.

%K nonn

%O 1,1

%A _Gus Wiseman_, May 15 2020